On a posteriori stopping rules of adaptive stochastic heavy ball method for ill-posed problems
For researchers solving large-scale ill-posed inverse problems, this work offers a computationally efficient stochastic method with theoretical convergence guarantees.
The paper develops an adaptive stochastic heavy ball method for ill-posed inverse problems, incorporating a momentum term and an a posteriori stopping rule based on the discrepancy principle that avoids computing all residuals at each iteration. Numerical experiments show efficiency for large-scale problems.
In this paper we develop a stochastic heavy ball method for solving ill-posed inverse problems. The method updates the iterate using only a randomly selected equation at each iteration step while incorporating a momentum term into the process. To facilitate fast convergence, we propose an adaptive strategy for selecting the step size and the momentum coefficient. Inspired by the spirit of the discrepancy principle, we introduce an {\it a posteriori} stopping rule for our adaptive stochastic heavy ball method. This rule avoids the need to compute residuals of all equations in the system at every iteration or at fixed frequency intervals, thereby enhancing computational efficiency and practicality. Additionally, convex penalty functions are employed to capture the specific features of the desired solutions. Under suitable conditions, we establish almost sure convergence as well as convergence in expectation. Extensive numerical experiments are conducted to evaluate the performance of the proposed method, demonstrating its efficiency and promising potential for solving large-scale ill-posed problems.